L(s) = 1 | + (0.5 − 0.866i)7-s + (−1.5 − 0.866i)13-s − 1.73i·19-s + (−0.5 − 0.866i)25-s + 37-s + (1 + 1.73i)43-s + (−0.499 − 0.866i)49-s + (−1.5 + 0.866i)61-s + (0.5 − 0.866i)67-s − 1.73i·73-s + (0.5 + 0.866i)79-s + (−1.5 + 0.866i)91-s + (1.5 − 0.866i)97-s + (1.5 + 0.866i)103-s − 2·109-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)7-s + (−1.5 − 0.866i)13-s − 1.73i·19-s + (−0.5 − 0.866i)25-s + 37-s + (1 + 1.73i)43-s + (−0.499 − 0.866i)49-s + (−1.5 + 0.866i)61-s + (0.5 − 0.866i)67-s − 1.73i·73-s + (0.5 + 0.866i)79-s + (−1.5 + 0.866i)91-s + (1.5 − 0.866i)97-s + (1.5 + 0.866i)103-s − 2·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.047281592\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.047281592\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.73iT - T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + 1.73iT - T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.165248785724656983582651011530, −8.006807662205299319200684831470, −7.61235372126608453422466007365, −6.85433948495754649695117689604, −5.90758592807487326334319707482, −4.75202940456689191078574519267, −4.51977472756465596439625184697, −3.12270504489969288711812826615, −2.28481665526116421541602402999, −0.71197459327781752028770392355,
1.73068103862767334495426726424, 2.47442362313862663710996778273, 3.72391015436283279827454019357, 4.64567970474956593744742090617, 5.46934323291334221500532674465, 6.12118354656560318744628813082, 7.23799822220650440798619854488, 7.78153581652768383801341344239, 8.642851317633414069579636949229, 9.420056535421049446472641110578